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This is a work in two parts devoted to solutions of the so-called {\em four Landau's problems} in Number Theory, listed by Edmund Landau at the 1912 International Congress of Mathematics. In Part I the {\em Goldbach's conjecture} is proved.…

Kuzmin-Landau Lemma basic for the bound of sum of exponentials is a nice example of the arithmetization of Analysis. Here we tell the interesting story of this Lemma emphasizing Landau's best constant.

The question of defining unique, generally applicable constrained second, and higher-order, derivatives is investigated. It is shown that second-order constrained derivatives obtained via two successive constrained differentiations provide…

We study some approximation problems on a strict subset of the circle by analytic functions of the Hardy space H2 of the unit disk (in C), whose modulus satisfy a pointwise constraint on the complentary part of the circle. Existence and…

Extremal Graph Theory is a very deep and wide area of modern combinatorics. It is very fast developing, and in this long but relatively short survey we select some of those results which either we feel very important in this field or which…

This is the second volume of a textbook for a two-semester course in mathematical analysis. This second volume is about analysis of multi-variable functions. The topics covered include Euclidean spaces, convergence of sequences, open sets…

We solve two continuous extremal problems on the classes of monotone functions: in the first problem we find extremal values for a line integral of a coordinate-wise monotone function of two variables from a rearrange\-ment-invariant class…

The main aim of this paper is to establish several Landau-type theorems for certain bounded poly-analytic functions and reduced poly-analytic functions that generalize some previously established results.

The classical Euler's problem on stationary configurations of elastic rod with fixed endpoints and tangents at the endpoints is considered as a left-invariant optimal control problem on the group of motions of a two-dimensional plane…

With this paper, we begin a series of studies of extremal problems for estimating distributions of martingale transforms of bounded martingales. The Bellman functions corresponding to such problems are pointwise minimal diagonally concave…

We propose the use of algebras of generalized functions for the analysis of certain highly singular problems in the calculus of variations. After a general study of extremal problems on open subsets of Euclidean space in this setting we…

The paper deals with the optimal control problem described by second order evolution differential inclusions; to this end first we use an auxiliary problem with second order discrete and discrete-approximate inclusions. Then applying…

Extremal Combinatorics is among the most active topics in Discrete Mathematics, dealing with problems that are often motivated by questions in other areas, including Theoretical Computer Science and Information Theory. This paper contains a…

This work advances knowledge of the threshold of prox-boundedness of a function; an important concern in the use of proximal point optimization algorithms and in determining the existence of the Moreau envelope of the function. In finite…

This is a survey of old and new problems and results in additive number theory.

Variable-exponent fractional models attract increasing attentions in various applications, while the rigorous analysis is far from well developed. This work provides general tools to address these models. Specifically, we first develop a…

We study the uniqueness of optimal solutions to extremal graph theory problems. Lovasz conjectured that every finite feasible set of subgraph density constraints can be extended further by a finite set of density constraints so that the…

This paper mainly addresses the extrema of a nonconvex functional with double-well potential in higher dimensions through the approach of nonlinear partial differential equations. Based on the canonical duality method, the corresponding…

We consider scalar equilibrium problems governed by a bifunction in a finite-dimensional framework. By using classical arguments in Convex Analysis, we show that under suitable generalized convexity assumptions imposed on the bifunction,…

This work deals with the Landau equation in a bounded domain with the Maxwell reflection condition on the boundary for any (possibly smoothly position dependent) accommodation coefficient and for the full range of interaction potentials,…